68 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



arrangement is therefore given in Table 8, 

 which omits the values of a not affording com- 

 parison and adds three interpolated values. 

 The general fact thus shown is that a increases 

 as fineness diminishes, but to this there are two 

 exceptions. The more important exception 

 is in grade (E), of which all values of a are less 

 than the corresponding values for either grade 

 (C) or grade (G). The values for grades (B) 

 and (C) appear to be about the same, although 

 the two grades differ notably in fineness. The 

 exceptions are associated with peculiarities of 

 the grades with respect to range of fineness. 

 In grade (E) the range for bulk fineness is about 

 twice as great as in any of the other grades; 

 and if this character indicates the cause of its 

 abnormally small values of a, then the abnor- 

 mally high values in grade (B) are explained 

 by its small range of fineness as compared to 

 the range of grade (C). 



TABLE 8. Values of a, from Table 7, arranged to show vari- 

 ation in relation to fineness of debris. 



The values of a inferred from the logarithmic 

 plots are not sufficiently precise to yield 

 quantitative laws of the relations of a to 

 fineness and the range of fineness, and it is 

 again necessary to seek instead the laws 

 governing competent slope. Does competent 

 slope vary inversely with fineness, and what 

 is its law? Does competent slope vary in- 

 versely with range of fineness, and what is its 

 law? 



Inasmuch as fine debris is moved by a rela- 

 tively slow current and as the force of the cur- 

 rent is a direct function of slope, the competent 

 slope for fine debris is less than for coarse. If 

 we accept the thesis of Leslie and Hopkins (see 

 p. 16) that competent bed velocity (V^) varies 

 with the sixth root of the volume or mass of 



the debris particle, then, since bulk fineness is 

 the reciprocal of volume, 



T ..A oc - 



If we assume, from the Chezy formula, that 

 mean velocity of current is proportional to the 

 square root of slope and apply it to component 

 mean velocity (V cm ) and competent slope 

 (S c ), we have 



I" oc fl - 5 

 ' cm "t 



If we further assume that bed velocity is pro- 

 portional to mean velocity, then, by combining 

 the three proportions and reducing, we obtain 



oc 



I 



F 2 - 33 " 



(17) 



As each of the three assumed laws is subject to 

 important qualifications, the product of their 

 combination must be regarded as but a rough 

 approximation to the law connecting com- 

 petent slope with fineness. 



Further light on the law is afforded by some 

 experiments made for the specific purpose of 

 determining competent slope. In these experi- 

 ments the discharge remained constant while the. 

 velocity was modified by changing the width of 

 the outfall. The slope of the bed had been pre- 

 pared in advance, and the slope of the water 

 surface was measured for each width of out- 

 fall. At each stage of the experiment the 

 movement of grains of debris along the bottom 

 was noted by such phrases as "many," "sev- 

 eral," "few," "very few," "none," the words 

 being used in that order as a sort of scale. 

 Competent slope was inferred from a compari- 

 son of these notes with the recorded slopes of 

 the water surface. The results are given in 

 Table 9. 



In a closely related series of experiments, 

 Table 10, a slope of debris was prepared in 

 advance, a small discharge was passed over it, 

 and the discharge was progressively increased, 

 with notes on the movement of debris grains. 

 These experiments gave competent discharge. 



In each series the depths were measured, and 

 from these the mean velocities (V m ) were 

 computed. 



The experimental determinations were indefi- 

 nite for several reasons. In the first place, 



